315 Introduction to String Field Theory Gary T. Horowitz Department of Physics University of California Santa Barbara CA 93106 316 1. INTRODUCTION There are several possible string field theories corresponding to the different types of strings. One can choose among bosonic, supersymmetric, or heterotic strings, and open or closed strings. In addition one has a choice between a gauge invariant or a gauge fixed formulation of the theory. At the present time, these theories are in various stages of development. In these lectures I will try to provide an introduction to string field theory. I have chosen to focus on a gauge invariant, bosonic open string field theory. This has the advantage of being general enough to illustrate many of the new features of string field theory, while being specific enough to keep the analysis simple. It is also one of the most developed theories at the moment. Even with this restriction, there are several actions which have been proposed. Much of our general discussion will apply equally well to all candidate theories, however when it is necessary to become more specific, we will focus on an approach due to Wit ten1 and the related purely cubic action2. In a sentence, string field theory is a second quantized field theory, of strings. To understand this better, it is perhaps instructive to compare string theory with ordinary point particle theories. Consider first a massless relativistic particle in flat spacetime. At the classical level, this is described by a worldline X^[T) extremizing the action: where g(r) is a one dimensional metric. This action is invariant under reparam- eterizations of r. The equation of motion (in the gauge g = constant) is simply = 0. The gauge invariance of the action leads to the constraint PpP» = 0 317 where the momentum is Pp = g_ i aXj»• . So the solutions are simply null geodesies. At the first quantized level, the particle is described by a wave function rp. ifr is initially a function on the space of configurations of the particle at one moment of r (which is just ordinary spacetime X*) and an external time T. These wave functions must satisfy Schroedinger's equation: and also the operator version of the constraint: P20 = -V2V> = 0 But H = P2, so the Hamiltonian is constrained to vanish. Thus Schroedinger's equation implies that i> is independent of T. This means that there is no explicit time dependence in ip. Time is already included in Xo. The constraint is son 2- times called the physical state condition. One also has the propagator giving the amplitude for the particle to propagate from one point in spacetime Xj* to another £ which can be expressed in terms of a path integral (ignoring gauge fixing): f |Xf >= J —SIX"*» over all paths starting at Xj1 and ending at X£ and over all metrics. (Since g is not dynamical it does not appear in the specification of states on the left hand side. Its value at the endpoints is integrated over.) At the second quantized level, the wave functions ^(XM) become quantum field operators 4>{X^) which create or destroy particles. The first quantized constraint equation becomes the second quantized linearized equation of motion: V V = 0 i 318 The non-linear terms in the equation of motion describe how several particles interact. They are not determined by the first quantized theory which of course describes a single particle. They are a new physical input which can be added subject to general principles such as renormalizability, gauge invariance, etc. j Now let us repeat the same three levels of description for a relativistic string in a flat spacetime. At the classical level the string is described by a worldsheet fl X (at T) extremizing the action: \ = \ I j where çaj is a metric on the worldsheet. This action is invariant under reparame- j terizations of a and r and also conformai rescalings of qaf,. The equation of motion 1 (in the conformai gauge çaj a f?a() is just the two dimensional wave equation: d2Xti = 0 The gauge invariance leads to the constraint Tab = 0 (1.1) where Tab = -^fr. tq At the first quantized level, the string is described by a wave function \li[X^{a)\ which is a function on the space of string configurations at one moment of world- sheet time r. Schroedinger's equation again implies that there is no explicit time dependence. However, ifr must satisfy the constraints: Ln\i{> > = 0 n > 0 (Lo-l)!^> = O (1.2) where Ln are the Fourier modes of the constraint (1.1) and are called the Virasoro operators. We cannot impose (1.2) for all n because of the well known anomaly 319 3 in the Virasoro algebra . The propagator giving the amplitude for a string in one configuration Xj*(tr) to propagate to X^(<J) can be expressed in terms of the functional integral (ignoring gauge fixing) < X»{o)\XÇ(o) >= J DX^Dq^c-W'^ where the integral is over all XM(cr, r) which start at X^(o) and end at X^ip) and over all metrics. This first quantized string is very conveniently described using BRST quantization. As we will see, in this approach one introduces anti- commuting ghost fields c and 6 and considers wave functions on the extended space 0[XM(0),C((T)]. There is an operator Q called BRST operator which has the convenient property that the infinite set of constraints (1.2) is replaced by the single constraint Q|V> >= 0. Finally, at the second quantized level, the wave function ip becomes a quantum string field operator A{X^(o),c(o)\ which creates or destroys entire strings. The linearized equation of motion is simply QA = 0. As before, one then adds interactions subject to general principles. The resulting theory is string field theory and the subject of these lectures. Having said what string field theory is, I must now say a few words about why it is needed. The fact is that most researchers in string theory today are not concerned with any of the three levels of string theory described above! Instead they work with a procedure for calculating string scattering amplitudes perturba- tively, using only first quantized methods, without the full machinery of string field theory.* Almost all discussions of conformai field theory, Riemann surfaces, and * This can also be done for point particle theories. However the resulting formalism is more cumbersome tha;. ordinary quantum field theory and so is usually avoided. One difference between this and string theory is that, for the point particle theories, the diagrams involved are not smooth one dimensional manifolds. 320 vertex operators fall in this category. Since these first quantized methods involve the beautiful geometry of Riemann surfaces and their moduli space, some people have suggested that this might be sufficient for a complete theory. However, this seems unsatisfactory for at least three reasons. First, this procedure is inherently perturbative: a Riemann surface of genus g corresponds to a gtlí order contribution to a scattering amplitude. Although one could in principle calculate to arbitrary order in perturbation theory, one still runs into problems. For example, there is no reason to expect the string coupling constant to be small in nature. In fact, various arguments4*5 indicate that the cou- pling constant is probably of order unity, which makes the perturbation expansion highly questionable. More seriously, one can show that the perturbation expan- sion must break down for any value of the coupling constant6. This is because an exact relation (partial wave unitarity) is violated by the tree level amplitudes for large values of the external momentum. Thus some higher loop amplitudes must become large to compensate. Another problem with an inherently pertur- bative procedure is that string theory may contain qualitatively new effects which can only be seen non-perturbatively. This is by now a familiar feature of ordi- nary quantum field theories. But most importantly, the main reason one wants a quantum theory of gravity is not to calculate graviton scattering perturbatively in fiat spacetime Rather, it is to gain a better understanding of the structure of spacetime at short distances, the origin of the universe, and the evaporation of black holes. In other words, one needs a better description of gravity in strong field regions where classical general relativity breaks down. These are certainly non-perturbative regions. The second reason these perturbative methods fall short of a complete theory is that they require a choice of background spacetime metric (and possibly other 321 fields) to perturb about. In the full theory these background fields should be dynamical. Although there is a way of selecting which backgrounds are classically allowed (by requiring conformai or superconformal invariance) there is no way of choosing among these classical alternatives. One needs a more fundamental description to discover solutions to the full quantum equations of motion. Finally, a more aesthetic objection is the following. One has the strong impres- sion that there is some fundamental principle underlying string theory analogous to the way differential geometry is the basis for general relativity or gauge invari- ance is the basis for QCD. It is difficult to discover this fundamental principle in a perturbative framework. One needs a more fundamental description. Of course there is no guarantee that string field theory is the right approach to these questions. We do not yet know the best logical framework for a fun- damental description of string theory. However field theory is the most familiar framework and quantum field theory has worked extremely well for theories with- out gravity e.g.